Joint estimation of precision matrices in heterogeneous populations

Abstract

We introduce a general framework for estimation of inverse covariance, or precision, matrices from heterogeneous populations. The proposed framework uses a Laplacian shrinkage penalty to encourage similarity among estimates from disparate, but related, subpopulations, while allowing for differences among matrices. We propose an efficient alternating direction method of multipliers (ADMM) algorithm for parameter estimation, as well as its extension for faster computation in high dimensions by thresholding the empirical covariance matrix to identify the joint block diagonal structure in the estimated precision matrices. We establish both variable selection and norm consistency of the proposed estimator for distributions with exponential or polynomial tails. Further, to extend the applicability of the method to the settings with unknown populations structure, we propose a Laplacian penalty based on hierarchical clustering, and discuss conditions under which this data-driven choice results in consistent estimation of precision matrices in heterogenous populations. Extensive numerical studies and applications to gene expression data from subtypes of cancer with distinct clinical outcomes indicate the potential advantages of the proposed method over existing approaches.

Consistent multiple changepoint estimation with fused Gaussian graphical models

“…The discussion in this paper studies the impact of implementing graph-wise smoothness constraints on precision matrix estimation through penalising edge variation with the group Frobenius norm. This contrasts with previous research that has predominently focussed on enforcing smoothness constraints at an edge-by-edge level [Kolar and Xing, 2011, Monti et al, 2014, Danaher et al, 2013, Saegusa and Shojaie, 2016. For instance, one may replace the group-fused penalty in (4) with a linearly seperable norm such as the 1 norm, i.e.…”

Testing for Pathway (in)Activation by Using Gaussian Graphical Models

“…Methods that focus on identification of local differences joint precision estimation in penalized fashion can be found in Guo et al (2011), Danaher et al (2014, Zhao et al (2014), Price et al (2015), Bilgrau et al (2015) and Saegusa and Shojaie (2016). Ha et al (2015) and Xia (2017) augmented such approaches by a post hoc test procedure for differential edge identification.…”

Nearly assumptionless screening for the mutually-exciting multivariate Hawkes process

“…For instance, in the case of the Gaussian graphical model, Luo, Song and Witten (2014) considered estimating the conditional dependence graph by screening the marginal covariances. In order for this procedure to have the sure screening property, one must make an assumption on the minimum marginal covariance associated with an edge in the graph, which is not required for variable selection consistency of penalized estimators (Cai, Liu and Luo, 2011; Luo, Song and Witten, 2014; Ravikumar et al, 2011; Saegusa and Shojaie, 2016). …”